Analysis of nonlinear frequency mixing in Timoshenko beams with a breathing crack using wavelet spectral finite element method
Introduction
Detection of structural damages at an early stage is crucial for ensuring integrity, safety, and reliability of structural components. Accordingly, the techniques of non-destructive evaluation and health-monitoring have been of continued interest in the past few decades. These techniques can be classified broadly into two categories, namely, the vibration-based (low frequency) and the wave-based (high-frequency) techniques. The former relies on appraising the alterations in the modal characteristics of the structural component caused by the damage, thereby facilitating its localization [[1], [2], [3], [4]]. While this method of health monitoring has received considerable attention, a reliable prediction of damage parameters (existence, location, intensity, type, etc.) is often jeopardized by the operational variabilities and sensitivity to the boundary conditions. In contrast, wave-based techniques, more specifically the guided wave-based techniques, offer a superior alternative primarily because of their ability to detect incipient damages and inspect a large area or the entire cross-sectional depth [[5], [6], [7]]. A large volume of literature expounds on the guided wave-based damage detection strategies both from the theoretical and experimental perspectives; a comprehensive review of which has been presented by Mitra and Gopalakrishnan [8].
Over the last couple of decades, there has been an increasing interest in exploiting the nonlinear interactions of guided waves with damages present in some of the ubiquitously found structural components, such as beams, rods, and plates [9]. Structural damage is considered to be a source of nonlinearity caused by various mechanisms operating at different length scales, such as stress-strain hysteresis [10], amplitude dependent dissipation [11], thermoplastic coupling [12], etc. Of particular interest is the nonlinearity caused by an intermittent contact between the two surfaces of damage upon the passage of an incident wave, a phenomenon referred to as breathing [13]. When an incident wave with central frequency f interacts with a breathing damage, the resulting spectrum shows additional frequency peaks at the integer multiples (super-harmonics) or the fractional multiples (sub-harmonics) of f, thus providing a precursor to the existence of damage [14]. Several researchers have elaborated an effective use of these pre-cursors in identifying and localizing a breathing damage in structural components [[15], [16], [17], [18], [19]].
In contrast to a single frequency excitation, a method based on dual frequency excitation has also been studied in the past. The basic variant of this method is referred to as the vibro-acoustic modulation (VAM) wherein the structural component is interrogated simultaneously by a low frequency (pumping) and a high frequency (probing) signal [20]. Upon interacting with a breathing damage, the probing signal gets modulated leading to the presence of sidebands in the frequency spectrum, a phenomenon referred to as the nonlinear frequency mixing. An operationally similar alternative to VAM is to interrogate the component with two high frequency signals with a small difference in their central frequencies [[21], [22], [23]]. In addition to retaining all essential features of VAM, this alternative eliminates the requirement of actuating the entire component with a low frequency signal (typically achieved using a vibration-bench) and facilitates the use of similar types of actuators (usually piezo-ceramic) used for excitation. In several of the previously reported studies dealing with the nonlinear frequency mixing, the focus has been on assessing the existence of such additional frequency peaks thereby confirming the presence of a breathing crack [[24], [25], [26], [27], [28]]. However, barring the recent investigations based on processing the time-domain signal [[29], [30], [31]], very few researchers have outlined schemes for localizing the breathing crack by using wave-based techniques; in particular, those involving the nonlinear wave-damage interactions.
In this paper, a method exploiting the dependency of the strengths of the frequency sidebands on the nature of the dual-frequency input waveform is outlined for localizing a breathing crack. The proposed method is based on the rationale that the strength of the frequency side-bands depends on the extent of temporal overlap that the constituent signals have, as they propagate through a breathing damage. Accordingly, when the two wave envelopes propagate simultaneously through the breathing damage (with 100% temporal overlap), maximum strength of the frequency side-bands is obtained and apparently, no side bands are observed when the two wave envelopes pass independently (without any temporal overlap) through a breathing damage. Furthermore, in a dispersive waveguide, the amount of temporal overlap between the two wave envelopes also varies as they advance along their direction of propagation. These observations are used constructively for devising an iterative damage localization scheme based on controlling the initial temporal overlap between the two constituent wave envelopes and thereafter monitoring quantitatively the strength of the frequency side-bands observed in the response spectrum. Because it is easy to tweak the temporal overlap between two wave-envelopes in the input signal, the proposed method lends itself to an operationally feasible alternative of localizing a breathing crack than the signal processing based methods suggested earlier [29,30].
To exposit the underlying steps involved in the proposed method, a case of Timoshenko beam with a transverse edge-crack is considered in the present analysis. A dual-frequency A0 wave, in the form of a tone-burst signal, is used as an excitation signal. An iterative use of the Wavelet Spectral Finite Element method (WSFE) is made for analyzing the propagation of elastic waves in presence of a breathing crack. The choice of WSFE, over the conventional Fourier Spectral Finite Element method (FSFE) [32], is driven by its amenability to analyze the finite length waveguides and it being devoid of the wrap-around effect typically associated with the use of FSFE [29,33]. A damage spectral element, incorporating the effect crack-induced flexibility, is employed in the finite element assembly. Using this numerical framework, the effect of temporal overlap between the two constituent wave envelopes on the strengths of the resulting frequency side-bands is brought out quantitatively. An iterative damage localization strategy is then outlined and its effectiveness is assessed out by considering a representative numerical example.
The remainder of this paper is organized in four sections. Section 2 opens with the definition of the problem under investigation together with the statement of mathematical quantities used in the ensuing analysis. Subsequently, an equivalent model employed for representing a breathing crack is described, followed by the statement and solution of the homogeneous form of the governing equation of motion. The obtained solutions are then used as the interpolation functions for formulating the healthy and damage wavelet spectral finite elements. An iterative procedure used for solving the resulting system of nonlinear equations is outlined subsequently. In Section 3, the frequency response spectrum of a Timoshenko beam with breathing crack, interrogated by a dual-frequency Lamb wave, is analyzed for the presence of higher harmonic as well as combination harmonic components. Further, the strength of both the aforestated frequency components is assessed as a function of the amount of temporal overlap between the two wave envelopes as they propagate through the breathing crack. Based on the inferences drawn in Section 3, a damage-localization strategy is proposed in Section 4. Summary and inferences drawn from the present investigation are included in Section 5.
Section snippets
Problem definition and the method of solution
In this section, the underlying problem of wave propagation through a cracked beam-type structure, modeled using the Timoshenko hypothesis, is described together with the definitions of various parameters used in the ensuing analysis. A configurationally similar problem, albeit based on the assumption of an open-crack, has been addressed by Krawczuk et al. [34] by employing the method of Fourier spectral finite elements. Because of the inherent assumption of periodicity in the solution, the use
Results and discussion
This section is organized in two subsections. In the former, response of the Timoshenko beam to a dual frequency excitation is examined. The constituent frequencies are selected from the group speed dispersion curve of the waveguide. The wave-damage interaction in case of a waveguide with open crack and that with a breathing crack is investigated in both time as well as frequency domains. In the latter half, influence of the temporal overlap between the two constituent frequencies on the
Crack localization
Fig.Ā 9 shows schematically the envelopes of the two wave signals at the crack location. The wave envelope with the higher central frequency f2 is assumed to arrive early in time at the crack location than that with f1. Further, tc2 and tc1 are the arrival times of the peaks of these two envelopes, respectively. As depicted in Fig.Ā 9, the temporal overlap between the two signals at the crack location, denoted by ĪtO, can be expressed asin which, sp1 and sp2 are the time
Summary and conclusion
In this paper, nonlinear frequency mixing resulting from the interaction of a dual-frequency A0 Lamb wave with a breathing edge-crack in Timoshenko beams is studied numerically. The response of the cracked waveguide to a dual-frequency tone-burst type excitation is analyzed in the setting of the WSFE method. It is demonstrated that the response spectrum at the location of the sensor is populated with several additional harmonic components in addition to the two fundamental frequency peaks
CRediT authorship contribution statement
D.M. Joglekar: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing - original draft, Writing - review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The author thankfully acknowledges the financial support from the Science and Engineering Research Board, Government of India through Grant No: ECR/2017/001171.
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